PHY 475/375 Lecture 10 (April 25, 2012) Multiple-Component Universes

So far, we have studied model universes that contain only one component — matter only, radiation only, etc. In this chapter, we will look at more sophisticated models containing two or more components.

We begin, as always, with the Friedmann equation; recall that it can be written in the form

2 2 8πG κc H(t) = 2 ε(t) 2 2 (6.1) 3c − R0 a(t) where H a/a˙ and ε(t) is the energy density contributed by all the components of the universe. ≡

Also, recall from an earlier lecture that equation (4.31) gave us a relation between κ, R0,H0, and Ω0: 2 κ H0 2 = 2 Ω0 1 (6.2) R0 c − Let us rewrite the Friedmann equation (6.1) without explicitly including the curvature by using equation (6.2): 8πG H2 H(t)2 = ε(t) 0 Ω 1 (6.3) 3c2 − a(t)2 0 −

2 Dividing by H0 , this becomes 2 H(t) ε(t) 1 Ω0 2 = + − 2 (6.4) H0 εc,0 a(t) where εc,0 is the critical density in the current epoch, given by 3c2H2 ε 0 (6.5) c,0 ≡ 8πG

Now, we know that our Universe contains matter, for which the energy density εm has the depen- 3 4 dence εm = εm,0/a , and radiation, for which the energy density εr has the dependence εr = εr,0/a . It may also have a cosmological constant, with energy density εΛ = εΛ,0 = constant. While it is certainly possible that the universe contains other components as well, we will consider only the contributions of matter (w = 0), radiation (w =1/3), and the cosmological constant Λ (w = 1). − Recall from an earlier lecture that the energy density ε is additive, as written in equation (5.4):

ε = εw w X PHY 475/375 (Spring 2012 ) Lecture 10

Using the additive property of the energy densities of diﬀerent components, together with the various dependencies on the scale factor written above, equation (6.4) may be written as

2 H εr + εm + εΛ 1 Ω0 2 = + −2 H0 εc,0 a

4 3 εr,0/a + εm,0/a + εΛ,0 1 Ω0 = + −2 εc,0 a

εr,0/εc,0 εm,0/εc,0 εΛ,0 1 Ω0 = 4 + 3 + + −2 a a εc,0 a

2 H Ωr,0 Ωm,0 1 Ω0 2 = 4 + 3 + ΩΛ,0 + −2 (6.6) ⇒ H0 a a a

where we have put Ωr,0 = εr,0/εc,0, Ωm,0 = εm,0/εc,0, ΩΛ,0 = εΛ,0/εc,0, and Ω0 = Ωr,0 + Ωm,0 + ΩΛ,0.

Note also that the Benchmark model introduced in an earlier lecture has Ω0 = 1 (i.e., spatially ﬂat), but while this is consistent with the observational data, it is not demanded by the data. Therefore, we will retain the curvature term (1 Ω )/a2 in equation (6.6). − 0 To proceed, we write H =a/a ˙ in equation (6.6), multiply both sides by a2, and take the square root; this gives Ω Ω 1/2 H−1 a˙ = r,0 + m,0 + a2 Ω + (1 Ω ) (6.7) 0 a2 a Λ,0 − 0

Integrating, we get the cosmic time t as a function of a: a da = H t (6.8) [Ω /a2 + Ω /a + a2 Ω + (1 Ω )]1/2 0 Z0 r,0 m,0 Λ,0 − 0

In general, this integral does not have a simple analytic solution. There are instances, however, when simple analytic approximations may be found. For example, we discussed in an earlier lecture how in a multi-component universe containing radiation, matter, and Λ, the radiation dominates in the early stages. With radiation dominating, and Ω0 = 1 for a ﬂat universe, equation (6.8) simpliﬁes to a da 1 a 1 a2 a H0 t = a da = ≈ Ω /a2 Ω Ω 2 Z0 r,0 r,0 Z0 r,0 0 p p p so that we get eventually a2 H0 t (6.9) ≈ 2 Ωr,0 p Page 2 of 12 PHY 475/375 (Spring 2012 ) Lecture 10

From equation (6.9), we get 1/2 a(t) 2 Ω H t (6.10) ≈ r,0 0 p

In the limit Ωr,0 = 1 (i.e., a spatially ﬂat universe containing only radiation), this becomes

1/2 a(t) 2 H t ≈ 0

Using t0 = 1/2H0 from equation (5.62) that we wrote in the last lecture for a ﬂat universe containing only radiation, this becomes

t 1/2 a(t) ≈ t 0 which matches equation (5.64) that we obtained for the scale factor in a spatially ﬂat universe containing only radiation.

Likewise, there will be epochs where matter dominates, or Λ dominates, as we discussed in an earlier lecture.

During some epochs, however, two of the components are of comparable density, and provide terms of roughly equal size in the Friedmann equation. During these epochs, a single-component model is a poor description of the universe, and we need to use a two-component model. We will now look at some of these instances in more detail.

First, we will examine a universe which is of historical interest to cosmology: a universe containing both matter and curvature (either negative or positive). In the years following Einstein’s revoking of Λ, cosmologists studied in detail the possibility of a spatially curved universe dominated by non-relativistic matter.

Matter + Curvature

In a curved universe containing only matter, we can put Ωr,0 =0, ΩΛ,0 = 0.

Also, since Ω0 = Ωr,0 + Ωm,0 + ΩΛ,0, having made the above choices (Ωr,0 =0, ΩΛ,0 = 0), we now have Ωm,0 = Ω0.

With the above choices, the Friedmann equation (6.6) in a curved, matter-dominated universe can be written in the form 2 H(t) Ω0 1 Ω0 2 = 3 + −2 (6.12) H0 a a

Note that although Ωm,0 = Ω0 is the only component, we retain the term (1 Ω0) for the curvature which means that Ω = 1; in other words, ε = ε . − 0 6 m,0 6 c,0 Page 3 of 12 PHY 475/375 (Spring 2012 ) Lecture 10

Let us now look at the characteristics of such a curved, matter-dominated universe. Suppose it is currently expanding (H0 > 0).

3 For it to stop expanding, we must have H0 = 0. Since Ω0/a is always positive, H(t)=0 • requires the second term on the right hand side of equation (6.12) to be negative. This means that a matter-dominated universe will cease to expand only if Ω0 > 1, and hence from equation (6.2), κ = +1 (positively curved).

We can ﬁnd the scale factor amax at maximum expansion by setting H(t) = 0 in equation • (6.12): Ω0 1 Ω0 0= 3 + −2 (6.13) amax amax so that a3 a2 max = max Ω Ω 1 0 0 − from which we get Ω a = 0 (6.14) max Ω 1 0 − Recall that Ω0 is the density parameter measured at a scale factor a(t0)=1. Also, note that in equation (6.12), the Hubble parameter enters only as H2, so there is a • time symmetry, and the contraction phase is just the time reversal of the expansion phase (although not at small scales, that is, small-scale processes will not be reversed during the contraction phase; e.g., stars will not absorb the photons they previously emitted). The eventual collapse of a Ω > 1 universe is sometimes called the “Big Crunch” — the • 0 universe returns to the hot, dense state in which it had originally started. In its contracting stage, an observer will see galaxies with a blueshift proportional to their distance, and wavelengths in the cosmic microwave background will compress progressively to shorter ones to eventually end in a cosmic γ-ray background.

What happens, however, in a matter-dominated universe with Ω < 1, corresponding to κ = 1 0 − (negatively curved)?

Both terms on the right hand side of equation (6.12) are then positive, so if such a universe • is expanding at t = t0, it will continue to expand forever (“Big Chill”).

At early times, when the scale factor is small (a Ω0/ 1 Ω0 ), the matter term in the • Friedmann equation (6.12) will dominate, so we can≪ write{ fro−m equation} (6.8) that

a a da a da a da a3/2 H t = = = 0 [Ω /a + (1 Ω )]1/2 [1/a + (1 Ω )/Ω ]1/2 ≈ [1/a]1/2 3/2 Z0 0 − 0 Z0 − 0 0 Z0 0 where we have used the fact that if a Ω0/ 1 Ω0 , then 1/a Ω0/ 1 Ω0 . From this expression, we see that the scale factor≪ will grow{ − at the} rate a ≫t2/3 at{ early− times.} ∝ Ultimately, the density of matter will be diluted far below the critical density, and the • universe will expand like the negatively curved empty universe we discussed in the last lecture, with a t. ∝ Page 4 of 12 PHY 475/375 (Spring 2012 ) Lecture 10

A plot of the scale factor a(t) vs. time, in units of H(t t0) is shown in Figure 6.1 of your text, and reproduced below. The panel on the right is an enlarged− view of the region enclosed by the small rectangle near “0” in the panel on the left.

The solid line shows the fate of a matter-dominated universe with Ω0 > 1 (corresponding to • κ = +1, positively curved). Such a universe expands until a maximum value of scale factor amax given by equation (6.14) and then collapses back down to a = 0 in a “Big Crunch.” The dashed line shows the perpetually expanding fate of a matter-dominated universe with • Ω < 1 (corresponding to κ = 1, negatively curved). At early times, the scale factor grows 0 − at the rate a t2/3, but ultimately when the density of matter falls far below the critical density, the scale∝ factor grows as a t. ∝ For perspective, the fate of a spatially ﬂat, matter-dominated universe (Ω =1, κ = 0) that • 0 we discussed in the last lecture is shown by the dotted line. Such a universe grows as a t2/3 for all time. ∝

Although the fate of the universe differs depending on whether Ω0 is greater than, less than, or equal to one, it is clear from the panel on the right above that it is very difficult at t t to tell ∼ 0 apart a universe with Ω0 slight less than one from one with Ω0 slightly greater than one; the right panel shows that the scale factors start to diverge significantly only after a Hubble time or more.

Matter + Lambda

Next, let us consider a universe which is spatially ﬂat, but which contains both matter and a cosmological constant Λ. Such a universe may be a close approximation to our Universe in the present epoch.

Spatial ﬂatness implies that Ωm,0 + ΩΛ,0 = Ω0 =1 so that Ω =1 Ω (6.22) Λ,0 − m,0

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The Friedmann equation (6.6) for a ﬂat universe containing matter and Λ then becomes

2 H Ωm,0 2 = 3 + ΩΛ,0 H0 a and using equation (6.22) can be written as

2 H Ωm,0 2 = 3 + 1 Ωm,0 (6.23) H0 a − The ﬁrst term on the right hand side of equation (6.23) represents the contribution of matter • and is always positive.

The second term on the right hand side of equation (6.23) represents the contribution of • a cosmological constant. It is positive if Ωm,0 < 1, implying ΩΛ,0 > 0, in which case the universe will continue to expand forever if it is expanding at t = t0, another example of a “Big Chill” universe.

On the other hand, the second term on the right hand side of equation (6.23) is negative if • Ωm,0 > 1, implying ΩΛ,0 < 0; such a negative cosmological constant provides an attractive force. A ﬂat universe with ΩΛ,0 < 0 will cease to expand at a maximum scale factor amax which can be found by setting H2 = 0 in equation (6.23):

Ω 0= m,0 + 1 Ω a3 − m,0 from which we obtain Ω 1/3 a = m,0 (6.24) max Ω 1 m,0 − Meanwhile, integration of the Friedmann equation (6.23) gives the solution

3/2 2 −1 a H0 t = sin (6.26) 3 Ω 1 amax m,0 − " # A plot of scale factor vs. time for thep above cases is shown in Figure 6.2 in your text, reproduced on the right. Just as for a positively curved, matter-only universe that we discussed previously, a flat universe with negative cosmological constant (ΩΛ,0 < 0) also ends in a “Big Crunch” (solid line), except that with the negative cosmological constant providing an attractive force, the lifetime of such a universe is ex- tremely short. Also shown in the figure is the “Big Chill” expansion of a Ωm,0 < 1, ΩΛ,0 > 0 universe (dashed line), and for perspective, the a t2/3 behavior (dotted line) of a spatially flat, matter-dominated universe (Ω =1, Ω = 0). ∝ m,0 Λ,0 Page 6 of 12 PHY 475/375 (Spring 2012 ) Lecture 10

It is worth doing a comparison of “Big Crunch” lifetimes between a positively curved matter-only universe and a ﬂat universe with a negative cosmological constant. While the graph in Figure 6.1 of your text (reproduced in this lecture on page 5) shows a “Big Crunch” after 110H−1 ≈ 0 in a positively curved universe containing only matter (Ωm,0 = Ω0 = 1.1), a ﬂat universe with a negative cosmological constant (Ω = 1.1, Ω = 0.1) has a lifetime of only 7H−1, as seen m,0 Λ,0 − ≈ 0 in Figure 6.2 of your text (reproduced on the previous page).

Of greater interest is a universe containing matter with a non-negative cosmological constant, which seems to resemble our Universe. In a ﬂat universe with Ωm,0 < 1 and ΩΛ,0 > 0, we discussed in an earlier lecture how the density contributions of matter and the cosmological constant are equal at the scale factor Ω 1/3 0.3 1/3 a = m,0 = =0.75 (6.27) mΛ Ω 0.7 Λ,0

With a ﬂat, ΩΛ,0 > 0 universe, the Friedmann equation (6.23) can be integrated to obtain

2 a 3/2 a 3 H0 t = ln + 1+ (6.28) 3 1 Ω  amΛ s amΛ  − m,0   The plot of a vs. t for such a “Bigp Chill” universe has already been discussed on the previous page.

At early times, when a a , equation (6.28) reduces to ≪ mΛ 2 a 3/2 H0 t = ln +1 3 1 Ω amΛ − m,0 " # which allows us to use the Taylor seriesp expansion: ln (1 + x)= x x2/2+ . . ., and retaining only the first term, we get − 2 a 3/2 H0 t ≈ 3 1 Ω amΛ − m,0 so that p 3/2 3 3/2 3 Ω 1/3 a(t)3/2 H t 1 Ω a = H t 1 Ω m,0 ≈ 2 0 − m,0 mΛ 2 0 − m,0 Ω " Λ,0 # p h i p and putting Ω =1 Ω , we get finally that at early times (a a ), Λ,0 − m,0 ≪ mΛ 3 2/3 a(t) Ω H t (6.29) ≈ 2 m,0 0 p with the a t2/3 dependence characteristic of a flat, matter-dominated universe. ∝ On the other hand, at late times when a a , equation (6.28) reduces to ≫ mΛ a(t) a exp 1 Ω H t (6.30) ≈ mΛ − m,0 0 which gives the a e(...)t dependence characteristicp of a flat, Λ-dominated universe. ∝ Page 7 of 12 PHY 475/375 (Spring 2012 ) Lecture 10

If we measure H0 and Ωm,0 in a ﬂat universe containing only matter and Λ, then the age of the universe can be found from equation (6.28) by putting a = a0 = 1:

−1 3/2 3 2H0 1 1 t0 = ln + 1+ 3 1 Ω  amΛ s amΛ  − m,0 p   3/2 3 2H−1 1 Ω 1/3 1 Ω 1/3 = 0 ln m,0 + 1+ m,0  − v −  3 1 Ωm,0 Ωm,0 ! u Ωm,0 ! − u  t  p   − 2H 1 1 Ω 1 = 0 ln − m,0 + 3 1 Ωm,0 s Ωm,0 Ωm,0 − " s # p −1 2H0 1 Ωm,0 +1 t0 = ln − (6.31) ⇒ 3 1 Ω Ω − m,0 "p m,0 # p p If we approximate our own Universe as having Ωm,0 =0.3 and ΩΛ,0 =0.7 (ignoring the contribution of radiation which, as we will see later has no signiﬁcant eﬀect on our estimate of t0), we get its current age to be t =0.964 H−1 = (13.5 1.3)Gyr (6.32) 0 0 ± assuming H = 70 7 km s−1 Mpc−1. 0 ±

The epoch at which Ωm = ΩΛ is obtained by putting a = amΛ in equation (6.28): −1 2H0 −1 t0 = ln 1+ √2 =0.702 H0 =9.8 1.0Gyr (6.33) 3 1 Ωm,0 ± − h i p Equation (6.33) implies that, if our Universe is well ﬁt by the Benchmark Model with Ωm,0 =0.3 and ΩΛ,0 0.7, then the cosmological constant has been the dominant component of our Universe for the last≈ 4 billion years or so.

Matter + Curvature + Lambda

In the above discussion, we showed that a flat universe with Ωm,0 > 1 and ΩΛ,0 < 0 is infinite in spatial extent, but has a finite duration in time.

On the other hand, we showed that a ﬂat universe with Ωm,0 1 and ΩΛ,0 0 extends to inﬁnity both in space and in time. ≤ ≥

So then, if a universe containing both matter and Λ is curved, a wide range of behaviors is possible for the function a(t). We can examine such behaviors by choosing diﬀerent values of Ωm,0 and ΩΛ,0 without constraining the universe to be ﬂat. Page 8 of 12 PHY 475/375 (Spring 2012 ) Lecture 10

Let us begin by writing the Friedmann equation for a curved universe containing both matter and a cosmological constant Λ: 2 H Ωm,0 1 Ωm,0 ΩΛ,0 2 = 3 + ΩΛ,0 + − 2 − (6.34) H0 a a Now, consider the following: If Ωm,0 > 0 and ΩΛ,0 > 0, then both the ﬁrst and second terms on the right hand side of • equation (6.34) are positive. If, however, Ω + Ω > 1, so that the universe is positively curved (based on equation • m,0 Λ,0 6.2), then the third term on the right hand side is negative.

2 As a result, for some choices of Ωm,0 and ΩΛ,0, the value of H will be positive for small values of a (where matter dominates) and for large values of a (where Λ dominates), but will be negative for intermediate values of a (where the curvature term dominates). Since negative values of H2 are unphysical, this means that those universes have a forbidden range of scale factors.

The possibilities stemming from the above discussion are best studied via the plot in Figure 6.3 in your text, which shows the general behavior of the scale factor a(t) as a function of Ωm,0 and ΩΛ,0. The plot is reproduced below.

The dashed line in the plot is for Ωm,0 +ΩΛ,0 = 1, and hence indicates flat universes with κ = 0. Models lying above this line have positive curvature (κ = +1), whereas models lying below this line have negative curvature (κ = 1). − In the region labeled “Big Crunch,” the uni- • verse starts with a = 0 at t = 0, reaches a maximum scale factor amax, then recol- lapses to a = 0 at a finite time t = tcrunch. Note that “Big Crunch” universes can be positively curved, negatively curved, or flat.

In the region labeled “Big Chill,” the uni- • verse starts with a = 0 at t = 0, then expands outward forever, with a as → ∞ t . Like “Big Crunch” universes, “Big Chill”→∞ universes can have any sign for their curvature.

In the region labeled “Big Bounce,” the universe starts out with a 1 and H0 < 0; that • is, it is contracting from a low-density, Λ-dominated state. As it contracts,≫ the negative curvature term in equation (6.34) becomes dominant, causing the contraction to stop at a minimum scale factor a = amin > 0 at some time tbounce. The universe then expands outward forever, with a as t . That is, it is possible to have a universe which expands outward at late→ times, ∞ but→ which ∞ never had an initial Big Bang (with a =0 at t = 0).

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Universes which fall just below the dividing line between “Big Bounce” and “Big Chill” • universes are “loitering” universes, sometimes also called “Lemaitre” universes. Such a universe starts in a matter-dominated state, expanding outward with a t2/3. Then, it ∝ enters the “loitering” stage, in which a is very nearly constant for a long period of time (almost like a static universe). The closer such a universe lies to the “Big Bounce” “Big − Chill” dividing line, the longer its loitering stage lasts. Eventually, however, the cosmological constant takes over, and the universe starts to expand exponentially.

Yet another way to study the diﬀerent types of expansion and contraction possible is by looking at a plot like Figure 6.4 in your text, which is reproduced below. The ﬁgure shows a(t) vs. time for four model universes. Each of these universes has the same current matter density parameter: Ωm,0 =0.3, measured at t = t0 and a = 1.

The four model universes depicted in this plot cannot be distinguished from each other by measuring their current matter density and Hubble constant. Yet, due to having different values for the cosmological constant, they have very different pasts and very different futures.

The dashed line in the ﬁgure above shows the scale factor for a universe with ΩΛ,0 = 0.3. • Since Ω + Ω = 0.3+( 0.3) = 0 < 1, this universe has negative curvature. As shown− m,0 Λ,0 − in the plot, it is destined to end in a “Big Crunch.”

The dotted line shows a(t) for a universe with ΩΛ,0 =0.7. Since Ωm,0 + ΩΛ,0 =0.3+0.7=1, • this universe is spatially ﬂat. As shown in the plot, it is destined to end in an exponentially expanding “Big Chill.”

The solid line shows a(t) for a universe with Ω =1.8. Since Ω +Ω =0.3+1.8=2.1 > • Λ,0 m,0 Λ,0 1, this universe is positively curved. It is a “Big Bounce” universe which contracted until a = a 0.56, then started expanding outward, which it will continue to do forever. bounce ≈

The dot-dash line shows a(t) for a universe with Ω =1.7134. Since Ω +Ω =0.3+1.7= • Λ,0 m,0 Λ,0 2.0 > 1, this universe is positively curved. It is a “loitering” universe, which starts out by expanding outward, but then spend a long period of time near a = aloiter 0.44. Eventually, it expands exponentially (the exponential part is barely visible in the scanned≈ ﬁgure, since it skims along the outer edge of the solid line; you should look at the ﬁgure in your text for better visibility).

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Since we are very close to accounting for all the components in our Universe, it is worth pausing to reﬂect whether any of the possibilities discussed above can be ruled out for our Universe. Strong observational evidence exists, in fact, to suggest that we do not live in a “loitering” or “Big Bounce” universe. If we lived in a “loitering” universe, then we would see nearly the same redshift for galaxies • with a very large range of distances. For instance, with a 0.44 (the appropriate loiter ≈ “loitering” scale factor for a universe with Ωm,0 = 0.3, such as in ours), there should be a large excess of galaxies at redshift

z =1/a 1 1/0.44 1 loiter loiter − ≈ −

that is, at zloiter 1.3. No such excess of galaxies is seen at this redshift, or indeed at any redshift, in our Universe.≈

If we lived in a “Big Bounce” universe, then as we looked out into space, we would see • redshifts increasing until a maximum zmax = 1/abounce 1, after which redshifts would decrease until they became actually became blueshifts. In− our Universe, we do not see very distant blueshifted galaxies.

The highest likelihood, at the moment, seems to be that our Universe is a “Big Chill” universe, fated to eternal expansion.

Radiation + Matter

In a previous lecture, we calculated that radiation-matter equality took place at a scale factor

Ωr,0 −4 arm 2.8 10 ≡ Ωm,0 ≈ × Near this scale factor, the universe is best described by a ﬂat model containing both radiation and matter.

The Friedmann equation around the time of radiation-matter equality is then

2 H Ωr,0 Ωm,0 2 = 4 + 3 (6.35) H0 a a After taking the square root, this may be rearranged as

1/2 1 a˙ Ω a Ω 1/2 Ω a 1/2 = r,0 1+ m,0 = r,0 1+ H a a4 Ω a2 a 0 r r,0 rm and put in the form − a da a 1/2 H0 dt = 1+ (6.36) Ω arm r,0 p Page 11 of 12 PHY 475/375 (Spring 2012 ) Lecture 10

Integrating equation (6.36), we get

2 1/2 4arm a a H0 t = 1 1 1+ (6.37) 3 Ω − − 2arm arm r,0 " # p In the limit a a (radiation-dominated phase), we get ≪ rm 1/2 a 2 Ω H t (6.38) ≈ r,0 0 p which matches the result for the radiation-dominated phase obtained in equation (6.10).

The time of radiation-matter equation (trm) can be found by setting a = arm in equation (6.37):

2 1/2 4arm 1 H0 trm = 1 1 1+1 3 Ωr,0 − − 2 p 4a2 1 = rm 1 √2 3 Ω − 2 r,0 p 4a2 1 = rm 1 3 Ω − √2 r,0 p so that, putting back arm = Ωr,0/Ωm,0, we get

3/2 Ωr,0 −1 trm 0.391 2 H0 (6.40) ≈ Ωm,0

For the Benchmark model, with Ω =8.4 10−5, Ω =0.3, we get r,0 × m,0 −5 3/2 (8.4 10 ) − − − t =0.391 × H 1 =3.34 10 6 H 1 rm (0.3)2 0 × 0

−1 and with H0 = 14 Gyr, the time of radiation-matter equality was

t =3.34 10−6 14 109 yr = 47, 000yr (6.41) rm × × So the epoch when the Universe was radiation-dominated was very brief. This explains why our estimate of the age of the Universe as 13.5 Gyr in one of the previous sections, ignoring the contribution of radiation, is reasonable. The minor correction to the estimate of age by including the eﬀects of radiation is drowned out by the 10% uncertainty in the value of H0.

Having looked at all these contributions, we are now ready to look at the contents, history, and future of our actual Universe.

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